Altitude Variability

The altitude in Colombia varies between 0 and 5775 meters above sea level: the highest place is located in Sierra Nevada de Santa Marta in the place known as Pico Colon (^{ArcGIS, 2018}). Figure 1 shows the map of altitude in Colombia.

Source: Agustín Codazzi Geographical Institute (^{Instituto Geográfico Agustín Codazzi, 2005})

Figure 1 Map of altitudes in Colombia 

Figure 1 illustrates the map of altitudes of Colombia and shows the differences in altitude throughout regions, which will modify the speed of sound; therefore, altitude is an important factor that has to be considered when calculating the speed of sound in Colombia.

Temperature Differences

In Colombia, the temperature is variable and depends on factors such as the altitude, resulting in diverse climates-values from -8 degrees Celsius up to approximately 45 degrees Celsius may be present; (^{Instituto de Hidrología, Meteorología y Estudios Ambientales, 2015}), (^{Hurtado M, 2015}). Figure 2 shows the variability in temperature throughout Colombia.

Source: Instituto de Hidrología, Meteorología y Estudios Ambientales (^{Instituto de Hidrología, Meteorología y Estudios Ambientales, 2011})

Figure 2 Map of Temperatures in Colombia 

This variability in temperature through the country also affects the speed of sound, so it will not be constant throughout the whole territory. Hence, we can affirm that the speed of sound always depends on the ambient temperature. (^{Hahn, Cruz, & Carvalho, 2019})

Relative Humidity

Conversely, relative humidity does not directly depend on altitude although it might be slightly related to the temperature. However, relative humidity does have a direct correlation to the surrounding vegetation or sources of water (IDEAM), so humidity will rise in a place close to a river, wetland or lagoon, or with rich vegetation, while somewhere with less water bodies will be less humid. Besides, the tropical zone is a region with low-pressure, which will also produce high relative humidity as well as storms.

According to the map shown in Figure 3, the relative humidity in Colombia varies depending on the place. The zones located South and West of the country (the regions of Amazonas, Caquetá, Putumayo, Vaupes, and Chocó) have more plants, trees, rivers, and wetlands and have values between 85 and 95% of relative humidity. On the other hand, zones located East of the country have higher density of forests and have a relative humidity that varies from 75 and 85% (the map shows these zones in green). Finally, the region called “La Guajira” is mostly desertic and has the lowest relative humidity values.

Source: Instituto de Hidrología, Meteorología y Estudios Ambientales (^{Instituto de Hidrología, Meteorología y Estudios Ambientales, 2011})

Figure 3 Map of relative humidity in Colombia 

Meteorological Stations in Colombia

The IDEAM has 39 meteorological stations, located in different places with different altitudes (0 - 3000 meters) throughout the country. Table 1 shows the list of each station and its altitude (the table lists only 38 stations; the station #39 is located in San Andres Island, which was not considered during this research).

Table 1 Meteorological stations in Colombia. 

Source: Adapted table from (^{Instituto de Hidrología, Meteorología y Estudios Ambientales, 2011})

Figure 4 shows each meteorological station located in the country. As stated before, each station is located at a different altitude, latitude, and longitude and stores temperature, relative humidity, altitude, atmospheric pression, and other data.

Source: Authors

Figure 4 Meteorological stations in Colombia 

Relationship Between Speed of Sound, Relative Humidity and Temperature

According to the information provided by these stations, there is a direct correlation between the relative humidity and temperature of a place and the speed of sound in said place. Figure 5 shows this behavior.

Source: Adapted from “High Accuracy Acoustic Relative Humidity Measurement in Duct Flow with Air” (^{Wilhelm van-Schaik, 2010})

Figure 5 Speed of sound relative to temperature and relative humidity 

These values are included in the article ‘High Accuracy Acoustic Relative Humidity Measurement in Duct Flow with Air’ (^{Wilhelm van-Schaik, 2010}) were performed in a laboratory controlling conditions of temperature and relative humidity. These results show that the higher the temperature and relative humidity, the higher the speed of sound.

Comparison Between Temperature as Dependent to Altitude

The International Standard Atmosphere (ISA) models the temperature profile for changes of temperature in the atmosphere (Figure 6). The behavior of temperature at tropospheric level is linear, and the temperature at the sea level is 15 °C for the ISA model.

Source: Adapted from “The International Standard Atmosphere (ISA)” (^{Mustafa Cavcar, 2008})

Figure 6 Altitude vs. Temperature 

This graph shows a behavior similar to that resulting from data from the NASA (model shown in Figure 7). The graph presents the temperature as function of the altitude for each layer of the atmosphere: troposphere, stratosphere, mesosphere, and thermosphere, although our study is focused only on the troposphere, i.e., between 0 and 10 km. According to data from NASA, there also exists a linear behavior that follows the equation (1) below (^{National Aeronautics and Space Administration, 2018}):

Where T is the temperature in degrees Celsius (°C), and ℎ is the altitude in meters (m).

Source: Adapted from NASA-Earth atmosphere model (^{National Aeronautics and Space Administration, 2018})

Figure 7 Altitude vs. Temperature. 

Applying equation (1) from NASA and using the IDEAM data regarding temperature [°C] and altitude [m] for each meteorological station, the trend shown in Figure 8 results.

Source: Authors

Figure 8 Temperature vs. Altitude. Comparison between NASA equation and data obtained from meteorological stations in Colombia 

Figure 8 shows the equation from NASA in a blue dashed line; the average temperature taken from each meteorological station in Colombia is represented with small circles, and the trend of these values is drawn as a continuous red line (the correlation coefficient r ² of 0.954 and uncertainty of linear terms was Δa = 1.92 x 10^-4 and Δb = 2.32 x 10^-1) Looking at the behavior of the equation from NASA and the resulting graph in Figure 6, it is possible to say that they are similar: in both cases the temperature at 0 m is approximately 15 °C. However, these results differ from the data taken from the meteorological stations in Colombia that establish 28,045 °C as the approximate temperature at 0 m. The equation obtained for temperature in function of the altitude with data measured in Colombia is:

Where T is the temperature in degrees Celsius (°C), and h is the altitude in meters (m). To calculate the speed of sound with dry-air conditions, the following equation 3 could be useful (^{Hass, 2017}):

However, there are no dry-air conditions in this case, so if there is a variation in relative humidity, Eliakim presents an equation that relates the relative humidity and temperature for determining the speed of sound (^{Eliakim, 2014}):

Figure 9 shows the map of speed of sound in Colombia using equation 4:

Source: Authors

Figure 9 Speed of sound in Colombia 

To obtain the map in the Figure 9, we used two parameters for determining the probabilities: the location of each meteorological station in the country with the data of speed of sound calculated in its place, and the total studied area of the country.

For estimating the speed of sound in Colombia, we used a Kernel Density Estimation (^{Thedoridis, 1998}). This non-parametric approach estimates the shape of a desired curve using the input data, this estimation applies the equation 5:

In equation 5, N is the number of meteorological stations, (x,y) is a vector with spatial coordinates (geographical location of each meteorological station), and K_H [U_i] is the gaussian kernel (^{Buhmann, 2000}); and (X,Y)(x,y) is the probability value of speed of sound at (x,y).

We used the ArcGIS program to calculate this probability. The uncertainty of the measure of altitude is 0.5m, temperature uncertainty is 0.05^oC, and humidity uncertainty is 0.5%. The speed of sound in Colombia obtained for the histogram ranges between 332 m/s and 348,80m/s, with a mean value of 346,16 m/s.

If the thermodynamical conditions do not vary significantly in the place of the study, it is possible to use the average of temperature and relative humidity when calculating the radius of lightning impact by applying equation 6:

Where time can be calculated like the interval between the observation of lightning and arriving sound to the place where the observer is. For example, this observer can be a high-speed camera with event recording features. If the camera can register sound, this option might be used for recording the arrival of the thunder; otherwise, it is possible to use an independent sound sensor that records events. Once the camera is set in place for the lightning event, the speed sensor will activate to record the arrival time of both the lightning bolt and the thunder sound; then, it is possible to calculate the radius of impact with this interval and the equation of speed of sound.

We used this calculated speed of sound in other research to determine the direction and distance of lightning impact in transmission towers.